The government also carried out renewed barrel bomb attacks on opposition-held districts, a day after bombardments that killed at least 24 people, the Syrian Observatory for Human Rights said.
The Britain-based monitor said loyalist troops had full control of the Leramun district after heavy clashes, and reported fighting for neighbouring Bani Zeid, which is also held by rebels.
The two areas have been used by rebels to launch rockets into government-held districts in the west of the city.
Aleppo has been roughly divided between government control in the west and rebel control in the east since mid-2012.
In recent weeks, regime advances around the city's outskirts have severed the only remaining route into the rebel-held eastern neighbourhoods, effectively placing them under siege.
Opposition forces have responded by firing barrages of missiles into government districts, killing scores of civilians.
"The importance of capturing Leramun and Bani Zeid is to stop the missile fire and also to further tighten the siege," said Observatory director Rami Abdel Rahman.
He said government forces had now surrounded Bani Zeid, reporting heavy air strikes in the area and ongoing clashes.
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The monitor had no immediate toll in the fighting.
Government forces effectively severed the opposition's Castello Road supply route on July 7, when they advanced to within firing range.
They have tightened the encirclement of the rebel-held east since then, taking parts of the road itself and prompting food shortages and spiralling prices in opposition neighbourhoods.
Syria's Al-Watan daily, which is close to the government, also reported advances in Leramun, an industrial area that once housed scores of factories.
And state news agency SANA carried a call from the military urging residents of eastern Aleppo to "join the national reconciliation and expel the foreign mercenaries" from their neighbourhoods.
The agency said the army had sent text messages to residents and fighters in the east identifying "safe passages" for civilians wishing to leave and urging rebels to lay down their weapons.
Government forces also continued to pound opposition areas of the city, with the Observatory reporting a new barrel bomb attack on Tuesday on the district of Al-Mashhad.
It said at least one child was killed and the toll was expected to rise because many people were still missing beneath the rubble.
On Monday at least 24 people, including five children and two women, were killed in a barrel bomb attack on the same neighbourhood, according to a new toll from the Observatory.
More than 280,000 people have been killed in Syria since the conflict began in March 2011 with anti-government protests that were met with a regime crackdown.